Realisation and discussion

Detector sensitivities, i.e. local slopes of the signal with respect to the position, were measured for a 536 nm (diameter) polystyrene sphere (see Fig. 2.9A).

Positions far from the centre of the trap are sampled less frequently resulting in increased statistical noise in the determination of detector sensitivities. This statistical error is inversely proportional to the square root of the number of data points. If desired, the accuracy could therefore be increased by longer measurement of the sphere’s position fluctuations.

Detector sensitivities along lateral directions (x, y) are maximal at the trap center (x = 0, y = 0) and decrease towards the rim of the trap. Along the axial direction (z), detector sensitivity decreases continuously from positions close to the geometrical focus (toward +z, compare Fig. 2.8A) by more than a factor of two. This behaviour of the detector sensitivities in both lateral and axial directions is in qualitative agreement with theoretical predictions for Rayleigh-scatterers by Pralle et al. (1999) and for arbitrary spheres in arbitrarily focused beams by Rohrbach et al. (2003).

The maximal detector sensitivity is larger along the x-axis (≈22 mV/nm) than along the y-axis (≈19 mV/nm), although both signals are identically amplified by the electronics. This is an effect of the linear polarization of the laser light, which introduces an asymmetry between xand y (see Rohrbach et al. (2003)).

Calibrated sphere positions can be reconstructed from measured QPD signals SX(t), SY(t) and SZ(t) using Equation (2.18). Position histograms and variances of the sphere’s thermal displacements were calculated (see Fig. 2.9B). As expected, variances of displacements are in-dependent of the sphere’s position in the optical trap, demonstrating proper signal calibration.

All calibrated position histograms have Gaussian profiles, indicating that it is appropriate to approximate an optical trap as a 3D harmonic potential. Interestingly, although the shape of the uncalibrated axial signal histogram (Fig. 2.9A, lower panel) is asymmetric, the calibrated positions have a symmetric Gaussian distribution. The distortion of the histogram of the uncalibrated SZ signals is caused by the strong increase in detection sensitivity towards the

Figure 2.9: Determination and correction of position detector nonlinearities. A) Detector sensitivities (circles) and signal histograms (crosses) computed according to Equation (2.17) from QPD signals that resulted from the Brownian motion of a microsphere in the optical trap. B) Variances of the sphere’s displacements (circles) and position frequency histograms (crosses) computed from the calibrated position traces. Gaussian profiles (solid lines) were fitted to position histograms. Experimental details: a polystyrene sphere (radius a=267 nm) was trapped in water at T ≈298 K (η_water ≈0.89·10⁻³ Ns/m²); signals were acquired for 5 s at 1 MHz; detector sensitivities and variances of the displacements were calculated at∆t= 5µs for motion alongxandy and at∆t= 50µs for motion alongz (compare to Fig. 2.8).

trap center (see again Fig. 2.9A, lower panel). Consequently, a calibration of the QPD’s SZ signal that ignores detector nonlinearities would result in an asymmetric position histogram.

This demonstrates that it is important to take detector nonlinearities into account in order to accurately determine the trapping forces as well as to avoid distortions in images obtained with Thermal Noise Imaging Microscopy.

2.4.3 Summary

A method was developed to calibrate the detection scheme that is used for tracking the probe position in TNIM (see section 2.3.2). The method employs the diffusion constant of the trapped particle as a local calibration standard. Using this method, local detector sensitivities can be measured. Moreover, the full detector response can be calculated from the measured sensitivities. Thus, calibration of position time traces becomes possible even for detector signals that depend on the particle’s position in a non-linear way. Importantly, the method works in situ, i.e. it does not require additional reference measurements but can be applied directly in the sample. This is particularly useful for applications where calibration measurements on fixed particles are not possible due to complex environments (e.g. in the interior of a polymer network, section 3.1). In particular, the calibration scheme developed was key for imaging of diffusive mobilities close to surfaces (section 3.2) and within the plasma membrane (section 4.4.3). By a simple calibration of the detector using one calibration factor for each axis (Gittes and Schmidt, 1997) particle mobilities would have appeared amplified or diminished by the nonlinear detector response.

3. 3D-TNIM:

Imaging constrained diffusion in 3D

Chapter outline

The main objective of this work was to use TNIM for investigation on constraints to diffusion of the transmembrane protein EGFR. However, TNIM does not only permit investigation of mobility and structure in membranes (2D-TNIM), but also within complex 3D samples (3D-TNIM). In contrast to 2D-TNIM, the marker sphere is not coupled to a membrane protein, but diffuses in 3D.

The 3D-TNIM measurements presented in this chapter validate the concept of the microscope and contain novel information on the diffusive mobility of microscopic particles in the pres-ence of constraints. In section 3.1, it is ascertained that the interaction of the probe with nano-structures can be detected. This is achieved by tracking thermal position fluctuations of a microsphere inside an agar gel. Agar gel consist of interconnected polymer filaments that hinder the diffusion of the sphere and thereby create inaccessible volumes.

In section 3.2, the measurement of diffusive mobilities in presence of objects that constrain the motion is established. The diffusive motion of a microsphere is observed in the vicin-ity of glass and polymer surfaces. Results are compared to existing theoretical predictions for hydrodynamic coupling of a sphere to a single rigid surface. Moreover, diffusive mobil-ities of microspheres could be measured in the direct vicinity to the surface of a living cell (section 3.2.5), being important for the interpretation of 2D-TNIM experiments where micro-spheres are coupled to membrane proteins.

3.1 Imaging 3D diffusion in a polymer network

The position of a micro-sphere in an optical trap fluctuates due to the thermal energy provided by surrounding solution molecules (see section 2.2). The TNIM allows one to measure these fluctuations with nanometer resolution over many seconds. In bulk solution, the position

fluctuations are restricted to a volume that is given by the optical trap (see section 2.2).

The basic idea of 3D-TNIM is that the presence of an object further restricts the movements of sphere. In this section, the concept of 3D-TNIM is validated by tracking the thermal position fluctuations of a microsphere inside an agar gel. Polymer filaments within the gel are expected impose 3D constraints on the diffusion of the sphere. For instance, a molecular thin polymer filament is expected to cause a steric depletion zone with at least one sphere’s diameter (Fig. 3.1A).